Zeek's grandaughter, Amber, discovers she is pregnant in the first episode of Season 6. Struggling with the news, she tells her mother second, who initially doesn't react well to the news, but eventually comes through. Amber then tells Ryan, Zeeks' father, who promises to be there for the baby.
Zeek is born in the episode, "We Made It Through The Night". He is named after his great grandfather, and is presumedly living with his mother and great grandmother in her home.
In a flash-forward scene, a toddler Zeek is shown coming home with Ryan. He is given to his mother and step-father, who also has a daughter of his own. His mother and father are not together, but are on good terms.